Maxwell–Boltzmann Statistics

Statistical mechanics provides a way to understand the macroscopic properties of a system (like temperature, pressure, and energy) based on the behavior of its microscopic constituents (like atoms and molecules). Maxwell-Boltzmann (MB) statistics is one of the fundamental pillars of this field, particularly useful for systems composed of distinguishable particles that do not interact strongly.

Maxwellian Velocity Distribution

The Maxwellian velocity distribution describes the probability of finding a particle in an ideal gas at a certain velocity at a given temperature. It's a continuous probability distribution that shows how the speeds of molecules in an ideal gas are distributed. The distribution is a function of temperature and the mass of the particles.

The distribution function, $f(v)$, is proportional to the number of particles having velocities between $v$ and $v + dv$. It's derived by considering the number of ways particles can be distributed among different energy states, assuming the particles are distinguishable and obey classical mechanics. The function is given by:

$f(v) = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}}$

Where:

• $f(v)$ is the velocity distribution function.
• $m$ is the mass of a single particle.
• $k_B$ is the Boltzmann constant ($1.38 \times 10^{-23}$ J/K).
• $T$ is the absolute temperature in Kelvin.
• $v$ is the speed of the particle.

This equation tells us that at low speeds, the probability increases with $v^2$ because there are more ways to achieve a certain speed with different velocity components. However, at high speeds, the exponential term $e^{-\frac{mv^2}{2k_B T}}$ dominates, rapidly decreasing the probability because higher kinetic energy is less likely. The peak of this distribution corresponds to the most probable velocity.

The shape of the Maxwellian distribution changes with temperature. As temperature increases, the distribution broadens, and the peak shifts towards higher velocities. This means that at higher temperatures, molecules move faster on average, and there is a wider range of speeds.

Mean, Root-Mean-Square, and Most Probable Velocities

From the Maxwellian velocity distribution, we can calculate several important characteristic velocities:

1. Most Probable Velocity ($v_p$)

This is the velocity at which the Maxwellian distribution function $f(v)$ is maximum. It represents the speed at which the largest number of particles are moving. To find it, we differentiate $f(v)$ with respect to $v$ and set the result to zero.

$v_p = \sqrt{\frac{2k_B T}{m}}$

This can also be expressed in terms of the molar mass ($M$) and the universal gas constant ($R = N_A k_B$, where $N_A$ is Avogadro's number):

$v_p = \sqrt{\frac{2RT}{M}}$

2. Mean Velocity ($v_{avg}$ or $\bar{v}$ or $v_m$)

This is the average speed of all the particles in the gas. It is calculated by integrating $v \cdot f(v)$ over all possible velocities.

$v_{avg} = \int_0^\infty v f(v) dv = \sqrt{\frac{8k_B T}{\pi m}}$

In terms of molar mass and universal gas constant:

$v_{avg} = \sqrt{\frac{8RT}{\pi M}}$

3. Root-Mean-Square Velocity ($v_{rms}$)

This is the square root of the average of the squares of the velocities of the particles. It is directly related to the kinetic energy of the gas molecules.

$v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\int_0^\infty v^2 f(v) dv}$

The calculation yields:

$v_{rms} = \sqrt{\frac{3k_B T}{m}}$

In terms of molar mass and universal gas constant:

$v_{rms} = \sqrt{\frac{3RT}{M}}$

It's important to note the relationship between these velocities: $v_p < v_{avg} < v_{rms}$.

Bose–Einstein Statistics

While Maxwell-Boltzmann statistics works well for distinguishable particles, many systems in nature involve indistinguishable particles, such as photons or atoms with integer spin (bosons). For these particles, we need quantum statistics. Bose-Einstein (BE) statistics applies to identical, indistinguishable particles that can occupy the same quantum state. These particles are called bosons.

Key principles of BE statistics:

• Particles are identical and indistinguishable.
• Particles can occupy the same quantum state.
• The order of particles does not matter (symmetric wave function).
• There is no limit to the number of particles in a given state.

Distribution Function

The Bose-Einstein distribution function, $n_{BE}(E)$, gives the average number of bosons occupying a quantum state with energy $E$ at a given temperature $T$.

$n_{BE}(E) = \frac{1}{e^{(E - \mu) / k_B T} - 1}$

Where:

• $E$ is the energy of the quantum state.
• $\mu$ is the chemical potential. For a system of bosons, $\mu$ must be less than the lowest energy state ($E_0$) to ensure $n_{BE}(E)$ is positive. Often, $\mu$ is taken to be zero for photons, as their number is not conserved.
• $k_B$ is the Boltzmann constant.
• $T$ is the absolute temperature.

Comparing this to the Maxwell-Boltzmann distribution, we see the key difference: the denominator has a '-1' for BE statistics, whereas MB statistics has no term in the denominator related to the number of particles in a state (implicitly assuming a very large number of states compared to particles, or distinguishable particles). This '-1' term allows multiple particles to occupy the same state.

At high energies (or high temperatures, where $E \gg \mu$), the exponential term becomes large, and the '-1' becomes negligible. In this limit, the Bose-Einstein distribution approaches the Maxwell-Boltzmann distribution:

$n_{BE}(E) \approx \frac{1}{e^{(E - \mu) / k_B T}} = e^{-(E - \mu) / k_B T}$ (for $E \gg \mu$)

This shows that for high energy states, the quantum nature of indistinguishable particles becomes less significant, and they behave more like classical particles.

Phonon Gas

In solid crystals, lattice vibrations can be quantized. These quantized vibrations are called phonons. Phonons behave as particles (bosons) and can be treated as a gas of quasiparticles. The energy of a phonon is related to its frequency ($\omega$) by $E = \hbar \omega$, where $\hbar$ is the reduced Planck constant.

The study of phonons is crucial for understanding properties like heat capacity and thermal conductivity in solids. The Bose-Einstein distribution is used to describe the occupation of phonon states.

At low temperatures, the phonon distribution shows that lower energy states are more populated. A key phenomenon related to phonons is Bose-Einstein condensation, although it's more directly observed with actual particles like ultracold atoms. However, the concept of collective excitations and their statistical behavior is central.

The specific heat of solids at low temperatures can be explained using the phonon gas model. Debye's model, for instance, treats the phonons as a gas and uses the BE distribution to calculate the internal energy and then the specific heat.

Black Body Radiation

A black body is an idealized object that absorbs all incident electromagnetic radiation and emits radiation based solely on its temperature. The spectrum of radiation emitted by a black body was a major puzzle in classical physics, leading to the development of quantum mechanics.

Max Planck, in 1900, solved this problem by proposing that energy is not emitted or absorbed continuously but in discrete packets called quanta. For electromagnetic radiation, these quanta are photons. The energy of a photon is given by $E = h\nu$, where $h$ is Planck's constant and $\nu$ is the frequency of the radiation.

The spectral radiance of a black body, which describes the intensity of radiation emitted at each frequency (or wavelength) as a function of temperature, is given by Planck's radiation law. This law can be derived using Bose-Einstein statistics applied to photons.

Planck's Law for the energy density per unit frequency interval is:

$u(\nu, T) = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu / k_B T} - 1}$

Where:

• $u(\nu, T)$ is the energy density per unit frequency interval.
• $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s).
• $\nu$ is the frequency.
• $c$ is the speed of light.
• $k_B$ is the Boltzmann constant.
• $T$ is the absolute temperature.

The term $\frac{1}{e^{h\nu / k_B T} - 1}$ is precisely the Bose-Einstein distribution function for photons, where the chemical potential $\mu$ is zero because the number of photons is not conserved (they can be created or destroyed, e.g., in emission and absorption processes).

Planck's law successfully explains the observed black body spectrum, resolving the "ultraviolet catastrophe" predicted by classical physics (which suggested infinite energy emission at high frequencies).

From Planck's law, we can derive other important laws:

• Wien's Displacement Law: $\lambda_{max} T = b$, where $\lambda_{max}$ is the wavelength at which the intensity is maximum, and $b$ is Wien's displacement constant ($2.898 \times 10^{-3}$ m·K). This shows that the peak of the spectrum shifts to shorter wavelengths (higher frequencies) as temperature increases.
• Stefan-Boltzmann Law: $P/A = \sigma T^4$, where $P/A$ is the power radiated per unit area, and $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W m^-2 K^-4). This states that the total energy radiated per unit surface area of a black body across all wavelengths is proportional to the fourth power of its absolute temperature.